We investigate the consistency strength of the statement: κ is weakly compact and there is no tree on κ with exactly κ+ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly κ+ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal κ the nonexistence of a tree on κ with exactly κ+ many branches and, in particular, the Perfect Subtree Property for κ, implies the consistency of ADℝ + DC.
Bibliographical noteFunding Information:
The second-listed author was supported by L’ORÉAL Austria, in collaboration with the Austrian UNESCO Commission and in cooperation with the Austrian Academy of Sciences — Fellowship Determinacy and Large Cardinals and Elise Richter grant number V844 of the FWF.
The first-listed author was supported by Austrian Science Fund (FWF) Lise Meitner grant 2650-N35.
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